Question

If \[ta{{n}^{-1}}a+ta{{n}^{-1}}b+ta{{n}^{-1}}c=\pi \], prove that a+ b + c = abc.

Answer 1

Hint: For solving this problem, first we apply the addition identity for the inverse of tan function to simplify the expression on the left hand side. After getting a simplified expression, we apply tan function to both sides. By using this methodology, we can easily obtain our answer.

Complete step-by-step answer:
First, we take left-hand side to simplify:
\[\Rightarrow ta{{n}^{-1}}a+ta{{n}^{-1}}b+ta{{n}^{-1}}c\]
Now, by applying the trigonometric property \[ta{{n}^{-1}}x+ta{{n}^{-1}}y={{\tan }^{-1}}\left( \dfrac{x+y}{1-xy} \right)\], we get
\[\Rightarrow ta{{n}^{-1}}\left( \dfrac{a+b}{1-ab} \right)+ta{{n}^{-1}}c\]
Applying the same formula again in the \[ta{{n}^{-1}}\left( \dfrac{a+b}{1-ab} \right)+ta{{n}^{-1}}c\] to simplify it further, we get
$\begin{align}
  & \Rightarrow ta{{n}^{-1}}\left( \dfrac{\dfrac{a+b}{1-ab}+c}{1-\left( \dfrac{a+b}{1-ab} \right)c} \right) \\
 & \Rightarrow ta{{n}^{-1}}\left( \dfrac{\dfrac{a+b+c-abc}{1-ab}}{\dfrac{1-ab-ac-bc}{1-ab}} \right) \\
 & \Rightarrow ta{{n}^{-1}}\left( \dfrac{a+b+c-abc}{1-ab-bc-ca} \right) \\
\end{align}$
Now, by using the right-hand side, we get
 ${{\tan }^{-1}}\left( \dfrac{a+b+c-abc}{1-ab-bc-ca} \right)=\pi $
Applying tan operation to both sides, we get
$\dfrac{a+b+c-abc}{1-ab-bc-ca}=\tan (\pi )$
As we know that the value of $\tan (\pi )$ is 0 and ${{\tan }^{-1}}\left( \tan \theta \right)=\theta $, so we simplified the above expression as,
$\dfrac{a+b+c-abc}{1-ab-bc-ca}=0$
Now, taking the denominator of the left-hand side to the right-hand side and multiplying it with 0. So, the right-hand side is 0. So, the remaining equation is
$\begin{align}
  & \Rightarrow a+b+c-abc=0 \\
 & \Rightarrow a+b+c=abc \\
\end{align}$
Hence, we proved the equivalence of both sides as required in the problem.

Note: The key concept involved in solving this problem is the knowledge of inverse properties related to tan function. Students must be careful while solving the fraction which involves a variety of terms. Silly mistakes are bound to occur in that particular section.